# How do you find all the asymptotes for function f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)?

Feb 26, 2016

$x = - \sqrt[5]{6}$ and $x = - \sqrt[5]{5}$ are two vertical asymptotes. Horizontal asymptote is $y = 11$.

#### Explanation:

To find asymptotes of f(x)=(11(x−5)^5*(x+6)^5)/(x^10+11x^5+30), we should factorize denominator $\left({x}^{10} + 11 {x}^{5} + 30\right)$

The factors of $\left({x}^{10} + 11 {x}^{5} + 30\right)$ are $\left({x}^{5} + 5\right)$ and $\left({x}^{5} + 6\right)$

As $\left(x + a\right)$ is a factor of $\left({x}^{5} + {a}^{5}\right)$ (as $- a$ is zero of latter)

two factors of $\left({x}^{10} + 11 {x}^{5} + 30\right)$ are $x = - \sqrt[5]{6}$ and $x = - \sqrt[5]{5}$

and hence $x = - \sqrt[5]{6}$ and $x = - \sqrt[5]{5}$ are two vertical asymptotes.

It is also apparent that the highest degree of numerator is $11 {x}^{10}$ and that of denominator is ${x}^{10}$. As their ratio is $11$

Horizontal asymptote is $y = 11$.